How to make a circular set of verticies square
After subdividing the cube/square to match the circle subdivisions number (with W-->Subdivide) you may try to join vertices using the Bridge Tool. Select the vertices with Alt+RMB as pictured below, hit Spacebar, type 'bridge edge loops' and then dissolve the newly created edgeloop. Select it with Alt+RMB, then hit X-->Dissolve Edges. At the very end select the vertices of a square, hit Spacebar, type Grid Fill and set up the settings as pictured below.